"Happy [is] the man who listens to wisdom." means "The man who listens to wisdom is happy." This is a common proverbial form and is usually used to give advice:
Wealthy is the man [...]
Healthy is the man [...]
Happy is the man [...]
This may be easier to understand if you use "will be" instead of "is":
Happy will be the man who listens to wisdom.
"[...] who changes Emma for Harriet" means "[...] who exchanges Emma for Harriet." To rewrite the entire sentence:
I mention no names; but the man who exchanges Emma for Harriet will be happy!
Since we don't have the entire context, I don't know if the exchange has already happened or if the speaker is suggesting that the man should exchange Emma for Harriet but the gist of the sentence is that whoever is courting Emma would be happier if they were courting Harriet.
Despite your tags, you asked how common the funny meaning in British English.
Hysterical would normally mean very funny in the more common uses in British English, although it does depend on the context. In common speech, I doubt most people would understand it to mean anything else, despite its history.
Affected by or deriving from wildly uncontrolled emotion
informal Extremely funny
That context is everything, however. If there's any hint of humour, then it's clearly one meaning; if there's the whiff of madness, it's another. Consider:
"I went to see the new Ben Stiller film - it was hysterical"
(with apologies, as this is clearly a ridiculous scenario, as anyone who's ever seen a Ben Stiller film would attest).
vs
"I saw Ben Stiller being restrained - he was hysterical"
vs
"I went to see Ben Stiller - he was hysterical"
The first of these would universally be understood to mean "it was very funny"; the second is clearly suggesting that he was emotionally out of control; the third is ambiguous, as it could mean either. You'd need to decide from the rest of the conversation.
There are historical, sexist connotations - the "women are hysterical" view (as per the womb comment and here) which may change understanding, dependent on the listener. Consider:
"My wife was hysterical"
vs
"My husband is hysterical"
Depending on their prejudices, people could be more likely to understand the former as "emotionally over the top" and the latter as "very funny". Welcome to the modern world.
EDIT
You're still concerned about this topic, so I'll address your questions very specifically - at least, from a UK perspective.
> How common is the usage of the term hysterical meaning “funny” in BrE and AmE?
Very. Most British people would understand hysterical to mean funny as a primary definition, despite what dictionaries say. US preference is hilarious for that use, although that sounds slightly sarcastic to UK ears.
> Does hysterical actually carry a negative connotation as suggested in the above extract, or does it only convey a neutral meaning?
Nothing negative about it (again, BrE). As per the sarcasm comment, it's perhaps even less negative than some other options.
> Is its usage currently on the downtrend?
I see no downtrend in common usage in the UK. Ngram suggests a slight decline in overall use of the word (i.e. whatever the meaning) in both BrE and AmE, but not significantly. Specific to the "funny" use, though, that's certainly not something that's born out in everyday speech in my experience.
Ngram UK
Ngram US
... with a slightly higher incidence in all-English fiction usage.
Best Answer
This is not a common idiomatic expression. It's likely that she meant it literally, and either it was a non-sequitur or you missed the context that would have explained it. For instance, if she had just mentioned some maintenance details about the hotel infrastructure, she could have been explaining that she pays attention to those kinds of details because of her husband's profession.